matita/contribs/CoRN-Decl/reals/IVT.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/reals/IVT".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: IVT.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  22
  23 include "reals/CPoly_Contin.ma".
  24
  25 (* UNEXPORTED
  26 Section Nested_Intervals
  27 *)
  28
  29 (*#* * Intermediate Value Theorem
  30
  31 ** Nested intervals
  32
  33 %\begin{convention}% Let [a,b:nat->IR] be sequences such that:
  34 - [a] is increasing;
  35 - [b] is decreasing;
  36 - [forall (i:nat), (a i) [<] (b i)];
  37 - for every positive real number [eps], there is an [i] such that [(b i) [<] (a i) [+]eps].
  38
  39 %\end{convention}%
  40 *)
  41
  42 alias id "a" = "cic:/CoRN/reals/IVT/Nested_Intervals/a.var".
  43
  44 alias id "b" = "cic:/CoRN/reals/IVT/Nested_Intervals/b.var".
  45
  46 alias id "a_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var".
  47
  48 alias id "b_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var".
  49
  50 alias id "a_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var".
  51
  52 alias id "b_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var".
  53
  54 inline "cic:/CoRN/reals/IVT/a_mon'.con".
  55
  56 inline "cic:/CoRN/reals/IVT/b_mon'.con".
  57
  58 inline "cic:/CoRN/reals/IVT/a_b'.con".
  59
  60 inline "cic:/CoRN/reals/IVT/intervals_cauchy.con".
  61
  62 (* begin hide *)
  63
  64 inline "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__".
  65
  66 (* end hide *)
  67
  68 inline "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con".
  69
  70 (*#* %\begin{convention}% Let [f] be a continuous real function.
  71 %\end{convention}%
  72 *)
  73
  74 alias id "f" = "cic:/CoRN/reals/IVT/Nested_Intervals/f.var".
  75
  76 alias id "f_contin" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var".
  77
  78 inline "cic:/CoRN/reals/IVT/f_contin_pos.con".
  79
  80 inline "cic:/CoRN/reals/IVT/f_contin_neg.con".
  81
  82 (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *)
  83
  84 alias id "f_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var".
  85
  86 alias id "f_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var".
  87
  88 inline "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con".
  89
  90 (* UNEXPORTED
  91 End Nested_Intervals
  92 *)
  93
  94 (* UNEXPORTED
  95 Section Bisection
  96 *)
  97
  98 (*#* ** Bissections *)
  99
 100 alias id "f" = "cic:/CoRN/reals/IVT/Bisection/f.var".
 101
 102 alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var".
 103
 104 alias id "a" = "cic:/CoRN/reals/IVT/Bisection/a.var".
 105
 106 alias id "b" = "cic:/CoRN/reals/IVT/Bisection/b.var".
 107
 108 alias id "a_b" = "cic:/CoRN/reals/IVT/Bisection/a_b.var".
 109
 110 alias id "f_a" = "cic:/CoRN/reals/IVT/Bisection/f_a.var".
 111
 112 alias id "f_b" = "cic:/CoRN/reals/IVT/Bisection/f_b.var".
 113
 114 (*#*
 115 %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ].
 116 %\end{convention}%
 117 *)
 118
 119 (* begin hide *)
 120
 121 inline "cic:/CoRN/reals/IVT/Bisection/Small.con" "Bisection__".
 122
 123 inline "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__".
 124
 125 inline "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__".
 126
 127 (* end hide *)
 128
 129 inline "cic:/CoRN/reals/IVT/a_lft.con".
 130
 131 inline "cic:/CoRN/reals/IVT/rht_b.con".
 132
 133 inline "cic:/CoRN/reals/IVT/lft_rht.con".
 134
 135 inline "cic:/CoRN/reals/IVT/smaller_lft.con".
 136
 137 inline "cic:/CoRN/reals/IVT/smaller_rht.con".
 138
 139 (* UNEXPORTED
 140 Hint Resolve smaller_lft smaller_rht: algebra.
 141 *)
 142
 143 inline "cic:/CoRN/reals/IVT/Cbisect'.con".
 144
 145 (* UNEXPORTED
 146 End Bisection
 147 *)
 148
 149 (* UNEXPORTED
 150 Section Bisect_Interval
 151 *)
 152
 153 alias id "f" = "cic:/CoRN/reals/IVT/Bisect_Interval/f.var".
 154
 155 alias id "C_f_apzero_interval" = "cic:/CoRN/reals/IVT/Bisect_Interval/C_f_apzero_interval.var".
 156
 157 (* begin hide *)
 158
 159 inline "cic:/CoRN/reals/IVT/Bisect_Interval/Small.con" "Bisect_Interval__".
 160
 161 (* end hide *)
 162
 163 inline "cic:/CoRN/reals/IVT/bisect_interval.ind".
 164
 165 inline "cic:/CoRN/reals/IVT/Cbisect_exists.con".
 166
 167 inline "cic:/CoRN/reals/IVT/bisect.con".
 168
 169 inline "cic:/CoRN/reals/IVT/bisect_prop.con".
 170
 171 (* UNEXPORTED
 172 End Bisect_Interval
 173 *)
 174
 175 (* UNEXPORTED
 176 Section IVT_Op
 177 *)
 178
 179 (*#* ** IVT for operations
 180 Same conventions as before.
 181 *)
 182
 183 alias id "f" = "cic:/CoRN/reals/IVT/IVT_Op/f.var".
 184
 185 alias id "f_contin" = "cic:/CoRN/reals/IVT/IVT_Op/f_contin.var".
 186
 187 alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var".
 188
 189 alias id "a" = "cic:/CoRN/reals/IVT/IVT_Op/a.var".
 190
 191 alias id "b" = "cic:/CoRN/reals/IVT/IVT_Op/b.var".
 192
 193 alias id "a_b" = "cic:/CoRN/reals/IVT/IVT_Op/a_b.var".
 194
 195 alias id "f_a" = "cic:/CoRN/reals/IVT/IVT_Op/f_a.var".
 196
 197 alias id "f_b" = "cic:/CoRN/reals/IVT/IVT_Op/f_b.var".
 198
 199 (* begin hide *)
 200
 201 inline "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__".
 202
 203 (* end hide *)
 204
 205 inline "cic:/CoRN/reals/IVT/interval_sequence.con".
 206
 207 inline "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__".
 208
 209 inline "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__".
 210
 211 inline "cic:/CoRN/reals/IVT/intervals_smaller.con".
 212
 213 inline "cic:/CoRN/reals/IVT/intervals_small''.con".
 214
 215 inline "cic:/CoRN/reals/IVT/intervals_small'.con".
 216
 217 inline "cic:/CoRN/reals/IVT/intervals_small.con".
 218
 219 inline "cic:/CoRN/reals/IVT/Civt_op.con".
 220
 221 (* UNEXPORTED
 222 End IVT_Op
 223 *)
 224
 225 (* UNEXPORTED
 226 Section IVT_Poly
 227 *)
 228
 229 (*#* ** IVT for polynomials *)
 230
 231 inline "cic:/CoRN/reals/IVT/Civt_poly.con".
 232
 233 (* UNEXPORTED
 234 End IVT_Poly
 235 *)
 236